Let \(z_0\)  be a repelling fixed point of \(f\)  with multiplier \(\lambda \) .
By the Kœnigs Linearization Theorem [1]},
there exists a holomorphic function \(l\)  defined in a neighbourhood of
0 such that \(l(0)=z_0\)  and locally, \(l^{-1}\circ f\circ l(z)=\lambda z\) .
It was observed already by Poincaré that \(l\)  continues to a holomorphic
function \(L\)  on the entire complex plane, meaning there exists an entire
map \(L\)  such that \(L(0)=z_0\)  and
\(f(L(z))=L(\lambda z)\) 
